Binary file handouts/ho06.pdf has changed

--- a/handouts/ho06.tex	Fri Oct 26 13:28:33 2018 +0100
+++ b/handouts/ho06.tex	Fri Oct 26 16:14:10 2018 +0100
@@ -785,18 +785,72 @@
 
 \subsubsection*{Implementing an Interpreter}
 
-%\bigskip
-%takes advantage of the full generality---have a look
-%what it produces if we call it with the string \texttt{abc}
-%
-%\begin{center}
-%\begin{tabular}{rcl}
-%input string & & output\medskip\\
-%\texttt{\Grid{abc}} & $\rightarrow$ & $\left\{((\texttt{\Grid{a}}, \texttt{\Grid{b}}), \texttt{\Grid{c}})\right\}$\\
-%\texttt{\Grid{bbc}} & $\rightarrow$ & $\left\{((\texttt{\Grid{b}}, \texttt{\Grid{b}}), \texttt{\Grid{c}})\right\}$\\
-%\texttt{\Grid{aac}} & $\rightarrow$ & $\varnothing$
-%\end{tabular}
-%\end{center}
+The first step before implementing an interpreter for fullblown
+language is to implement a simple calculator for arithmetic
+expressions. Suppose our arithmetic expressions are given by the
+grammar:
+
+\begin{plstx}[margin=3cm,one per line]
+: \meta{E} ::= 
+   | \meta{E} \cdot + \cdot \meta{E} 
+   | \meta{E} \cdot - \cdot \meta{E} 
+   | \meta{E} \cdot * \cdot \meta{E} 
+   | ( \cdot \meta{E} \cdot )
+   | Number \\
+\end{plstx}
+
+\noindent
+Naturally we want to implement the grammar in such a way that we can
+calculate what the result of \texttt{4*2+3} is---we are interested in
+an \texttt{Int} rather than a string. This means every component
+parser needs to have as output type \texttt{Int} and when we assemble
+the intermediate results, strings like \texttt{"+"}, \texttt{"*"} and
+so on, need to be translated into the appropriate Scala operation.
+Being inspired by the parser for well-nested parentheses and ignoring
+the fact that we want $*$ to take precedence, we might write something
+like
+
+\begin{center}
+\begin{lstlisting}[language=Scala,basicstyle=\small\ttfamily, numbers=none]
+lazy val E: Parser[String, Int] = 
+  (E ~ "+" ~ E ==> { case ((x, y), z) => x + z} |
+   E ~ "-" ~ E ==> { case ((x, y), z) => x - z} |
+   E ~ "*" ~ E ==> { case ((x, y), z) => x * z} |
+   "(" ~ E ~ ")" ==> { case ((x, y), z) => y} |
+   NumParserInt)
+\end{lstlisting}
+\end{center}
+
+\noindent
+Consider again carfully how the semantic actions pick out the correct
+arguments. In case of plus, we need \texttt{x} and \texttt{z}, because
+they correspond to the results of the parser \texttt{E}. We can just
+add \texttt{x + z} in order to obtain \texttt{Int} because the output
+type of \texttt{E} is \texttt{Int}.  Similarly with subtraction and
+multiplication. In contrast in the fourth clause we need to return
+\texttt{y}, because it is the result enclosed inside the parentheses.
+
+So far so good. The problem arises when we try to call \pcode{parse_all} with the
+expression \texttt{"1+2+3"}. Lets try it
+
+\begin{center}
+\begin{lstlisting}[language=Scala,basicstyle=\small\ttfamily, numbers=none]
+E.parse_all("1+2+3")
+\end{lstlisting}
+\end{center}
+
+\noindent
+\ldots and we wait and wait and \ldots wait. What is the problem? Actually,
+the parser just fell into an infinite loop. The reason is that the above
+grammar is left-recursive and recall that parser combinator cannot deal with
+such grammars. Luckily every left-recursive context-free grammar can be
+transformed into a non-left-recursive grammars that still recognise the
+same strings. This allows us to design the following grammar
+
+
+
+
+